SOME PROBLEMS IN ESTIMATING POPULATION SIZES 

 FROM CATCH-ATAGE DATA 



Roy Mendelssohn' 



ABSTRACT 



A new method for estimating population sizes from catch-at-age data is given. The method treats the 

 observed population sizes as missing data and uses a combination of the Kalman filter and the EM 

 algorithm to derive maximum likelihood estimates of the parameters and minimum mean square error 

 estimates of the population sizes. The algorithm does not assume that the observation errors and the 

 errors in the population dynamics are uncorrected with equal variances, which is a common assumption 

 of existing techniques. A new parameterization for both recruitment and fishing mortality is given, based 

 on smoothness priors. Recruitment (or fishing mortality) is estimated as a nonparametric function of 

 time by calculating an "optimal" tradeoff between goodness-of-fit and smoothness of the function. The 

 algorithm allows for multiple sources of observations (fishing, surveys, etc.) and allows for missing data 

 in the observations, which can arise if the different sources of the observations occur on different time 

 scales. An example suggests that the new algorithm may better capture variation that is important when 

 using the population estimates to study the role of the environment (or other exogenous variables) on 

 the population dynamics. 



I can address the motivation of this paper by con- 

 sidering a slightly modified version of a model pro- 

 posed by Colhe and Sissenwine (1983). Assume that 

 the underlying population dynamics satisfy 



A^(a + 1, ^ + 1) = [Nia,t) - C(a,0] m + w{a,t) 



a 



1,A 



(1) 



where N{a,t) is the number of fish age a at time t, 

 C{a,t) is the catch of age a fish at time t, m = 

 exp{-m) is the mortality rate and the vector w{t) 

 = iw{l,t), . . .,w{A,t)y is a sequence of indepen- 

 dent, identically distributed normal random vectors 

 with mean and covariance matrix Q, and for any 

 vector a, the notation a^ denotes the transpose of 

 the vector. We assume that the initial population 

 vector A'^(O) is gaussian with a mean of pi and 

 covariance 1. 



The population itself is not observed. Instead we 

 observe that 



n{a,t) = q(t)Nia,t) + v{a,t) (2) 



where q is an unknown parameter and the vector 



'Southwest Fisheries Center Pacific Fisheries Environmental 

 Group, National Marine Fisheries Service, NOAA, P.O. Box 831, 

 Monterey, CA 93942. 



v{t) = (v(l,t), .. .,v (A, 0)^ is a sequence of indepen- 

 dent, identically distributed normal random vectors 

 with mean and covariance matrix R. It is assumed 

 that E{v{t) w{ty) = 0; that is, the observation 

 error and the underlying randomness in the popula- 

 tion are uncorrelated. There is some interest in the 

 value of the estimates of the q{t) (or if mortality is 

 to be estimated, in the estimate of m) but the major 

 interest lies in estimating the unobserved popula- 

 tion sizes N{a,t). The estimates of the N{a,t) should 

 reflect not only the trend in the population, much 

 as a regression might, but also the period-to-period 

 variation of the population, such as might be related 

 to environmental changes. This will be the emphasis 

 throughout the paper. 



The model described in Equations (1) and (2) dif- 

 fers from that of Collie and Sissenwine (1983) in that 

 I do not assume that the mortality rate is known; 

 here I allow the underlying population dynamics to 

 be random, and the observation errors in Equation 

 (2) to be additive rather than multiplicative. For 

 known in, Collie and Sissenwine (1983) suggested 

 minimizing 



T A 



1 1 (v(a,0' + wia,tf) 



t=\ a=l 



(3) 



Manuscript accepted July 1988. 



FISHERY BULLETIN: VOL. 86, NO. 4, 1988. 



over the parameters (in our notation) 9 = (q,N{t)) 



617 



